Sample Final Exam 4 MATH 1110 CALCULUS I FOR ENGINEERS

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1 Dept. of Math. Sciences, UAEU Sample Final Eam Fall 006 Sample Final Eam MATH 0 CALCULUS I FOR ENGINEERS Section I: Multiple Choice Problems [0% of Total Final Mark, distributed equally] No partial credit on this part, but show all your work anyway. It might help you if you come close to a border. Circle only ONE (the correct answer) for each problem. - Estimate the intervals where the function shown below is concave up and/or concave down. A) concave up for > 0, concave down for < 0 B) concave up for < <, concave down for <, > concave up for < <, concave down for <, > D) concave up for < < 0, >, concave down for <, 0 < < - The unit price of units of a certain product is given by 0 P ( ) =. What is + the maimum possible revenue when selling units? Round answer to nearest dollar. A) $8 B) $0 $ D) $7 /8

2 Dept. of Math. Sciences, UAEU Sample Final Eam Fall Find the function f ( ) satisfying the given conditions: f ''( ) =, f '(0) =, f(0) = 9 A) 9 B) D) - Identify the graph and the area bounded by the curves y = + cos and y = sin on the interval. A) area = + B) area = area = + D) area = /8

3 Dept. of Math. Sciences, UAEU Sample Final Eam Fall Identify the mass of a thin wire with density ( ) = + sin, A) 7 6 B) D) 7 6 ρ 6- Evaluate the integral. / 0 sec e tan tan A) e + c B) e D) e 7- Find an antiderivative by reversing the chain rule, product rule or quotient rule. A) B) D) cos sin sin cos cos 8- Which substitution can be used to integrate 9? A) = tanθ B) = secθ = sinθ D) = sinθ /8

4 Dept. of Math. Sciences, UAEU Sample Final Eam Fall 006 Section II [60% of Total Final Marks, distributed equally] Show all work (no work no credit). Write clearly. - Find the derivatives of the following functions (DON T SIMPLIFY): sec a) f ( ) = sin ( ) + b) g( t) = e t tan t ln t t c) y ) = ln ( 8, using properties of logarithms where needed. + - Find an equation of the tangent line to y = f() at =, f ( ) = The total cost of producing and marketing number of units of a certain product is given by C ( ) =. For what number is the total cost a minimum? Round answer to nearest unit. - Find the following integrals a) + b) 0 cos sin / c) tan sec - Find the area between the following curves on the given interval. y = e, y = e +, 0 6- Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by about = 7. y= and y= What is the maimum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 0 feet? [The density of water is 6. lbs/ft.] /8

5 Dept. of Math. Sciences, UAEU Sample Final Eam Fall Determine the following significant features of necessary, and sketch the graph. a). intercepts b). asymptotes c). etrema d). inflection points f( ) = +, approimating if /8

6 Dept. of Math. Sciences, UAEU Sample Final Eam Fall 006 Section III: [0% of Total Final Marks, distributed equally] - Sketch a graph of a function f ( ) corresponding to the given graph of y = f '( ). y y /8

7 Dept. of Math. Sciences, UAEU Sample Final Eam Fall Write the epression as a single integral: f ( ) f( ) A) f ( ) B) f ( ) D) ( ) f f ( ) - Use the graph to determine whether f ( ) is positive or negative. 0 y Make the indicated substitution for an unspecified function f ( ). / u = sin for ( cos ) f ( sin ) / / / sin ( ) du B) A) ( f u ) f ( u) du cos( ( )) f u du D) / / f ( u) du 7/8

8 Dept. of Math. Sciences, UAEU Sample Final Eam Fall You are given a pair of integrals, below. Evaluate the integral that can be worked using the techniques covered so far (the other cannot) and A) 9tan + c B) 9 tan + c 7 tan + c D) 9 tan + c 6- Find an antiderivative of the function 6 BONUS (OPTIONAL % of the Total Mark) SOLVE ONLY ONE OF THE FOLLOWING Is it true that the concavity of the graph of a twice differentiable function y=f() Changes every time f ( ) = 0? Give reasons for your answer. OR An advertisement consists of a rectangular printed region plus in. margins on the both sides and in. margins at top and bottom. If the area of the printed region is to be 9 in, find the dimensions of the ptinted region overall advertisement that minimize the total area 8/8